Finite index subgroups without unique product in graphical small cancellation groups

TL;DR

This paper constructs specific hyperbolic groups lacking the unique product property and demonstrates that their finite index subgroups can also lack this property, extending previous constructions to graphical small cancellation groups.

Contribution

It generalizes Comerford's construction to graphical small cancellation groups, showing subgroups can also lack the unique product property.

Findings

Constructed torsion-free hyperbolic groups without unique product

Showed subgroups up to finite index can also lack unique product

Extended graphical small cancellation techniques to subgroup constructions

Abstract

We construct torsion-free hyperbolic groups without unique product whose subgroups up to some given finite index are themselves non-unique product groups. This is achieved by generalising a construction of Comerford to graphical small cancellation presentations, showing that for every subgroup $H$ of a graphical small cancellation group there exists a free group $F$ such that $H*F$ admits a graphical small cancellation presentation.